flow

The method of taking notes in cross examination debate. It consists of 8 columns of space on a piece of paper where notes from each speech are taken or flowed. The purpose is to put rebuttals to arguments next to the original argument so that you can easily see the flow of the debate round and see what arguments to respond to.

In graph theory, flow is defined on a capacitated network as an assignation of a flow number to each arc such that the flow of each arc is less than or equal to that arc's capacity, and such that the flow into each vertex (besides the source(s) and sink(s)) is equal to the flow out of that vertex. The second property is called conservation of flow. The total flow is defined as the flow out of the source (= the flow into the sink).

Basically, you've got a network of pipes with junctions between them, one junction where all the flowing stuff comes from, and another junction where all the stuff ends up. Each pipe has a limited capacity, and can only move stuff in one direction (one-way valves?). You can send your stuff through the pipes any way you like, so long as you don't overload any of them. One interesting problem is to find the maximum flow through a network.

An element of the skill of an MC or rapper. Sometimes it is a verb (e.g. "That guy can flow"), sometimes a noun ("The boy's got flows"), often in non-standard English. While to some extent the flow will be dependent on the lyrics, it is more closely related to the execution. Cadence or rhythm may be appropriate synonyms.
Some descriptions of this talent take place as metacommentary within rap/hip-hop songs themselves, usually relying heavily on simile and metaphor to highlight certain elements. For example:

Two pages cause panic, freestyles is frantic
Thoughts is organic, flow is aerodynamic
Mic is volcanic, rhymes spread across the planet
I send out the scribe now the vibes gigantic
Now, I'm internationally known, mental capacities blown
I hope your microphone's accident prone
Raw footage is shown, but only showin scenes of my own
Leavin your periphereal vision in a zone
But not the twilight, from the insight that I write
and recite, in my mic'll be bright, cause I like
the night glowin, it's out of sight when I'm showin
Man I keep it goin, damn I keep it flowin

Other artists might differ, but a couple of things stand out from this verse: flow is original ("only showin scenes of my own"), smooth ("aerodynamic"), "organic", but yet overpowering, explosive ("volcanic"). To some extent, it makes new things visible: the mic becomes "brighter" through its exercise.

These people know how to make and break their own rules within verses and songs. While avoiding both mechanical attachment to the rhythm of the song and any disturbing errors in rhyme or length, they manage to feel like an organic part of the track. One important indicator is that, if you listen closely, you feel that if you knew lyrics, you could rap the lyrics like that and it would still sound pretty good, because they have found a flow that cannot help but match the track. It's infectious, you find yourself saying the lines like they say them. When he or she ends a verse, you feel like the verse needed to end there, not only from the logical structure of the lyrics but the trajectory of accent and emphasis that preceded it. When a rapper flows well, it feels like a well-known drinking song as opposed to a first-time slam poet.

The difficulty of this concept is obvious from the number of tracks with awful flow that even artists esteemed for their flow have. Compare Nas's Illmatic (where, in my opinion, all the tracks are at least good, B+/A-) to It Was Written or I Am.... More than the commercialism, bling, and lyrics, he just doesn't pull it off quite like he does on Illmatic. The lines are disjointed, and he feels disconnected from the track as a whole.

I don't rap, so I couldn't tell you how to do it. I merely present this as a point of departure for criticism of various MCs. Talib Kweli, for instance, sounds like each track he's written is just too short for all the things he trying to say. It's the reason why I can't listen to more then 1 or 2 songs at a time of many otherwise highly esteemed artists. For some reason or another, their ability to "flow" is inconsistent or nonexistent.